$\Pi_1^0$-ordinal analysis beyond first-order arithmetic

Abstract

In this paper we give an overview of an essential part of a $\Pi^0_1$ ordinal analysis of Peano Arithmetic (PA) as presented by Beklemishev ([2]). This analysis is mainly performed within the polymodal provability logic $\glp_\omega$.

We reflect on ways of extending this analysis beyond PA. A main difficulty in this is to find proper generalizations of the so-called Reduction Property. The Reduction Property relates reflection principles to reflection rules.

In this paper we prove a result that simplifies the reflection rules. Moreover, we see that for an ordinal analysis the full Reduction Property is not needed. This latter observation is also seen to open up ways for applications of ordinal analysis relative to some strong base theory.